Minimizing L 1 over L 2 norms on the gradient.
Chao Wang1,2, Min Tao3, Chen-Nee Chuah2
1Department of Statistic and Data Science, Southern University of Science and Technology, Shenzhen 518055, China.
Inverse Problems
|July 11, 2022
Summary
This study introduces L1/L2 minimization on image gradients, outperforming traditional L1 total variation for sparsity. This method enhances image recovery in MRI, CT, and low-frequency measurements.
Area of Science:
- Image processing
- Computational imaging
- Optimization theory
Background:
- Sparsity promotion is crucial for image reconstruction.
- L1/L2 norm is a superior approximation of L0 norm compared to L1 norm.
- Total variation (L1 norm on gradient) is a standard for image gradient sparsity.
Purpose of the Study:
- To investigate the efficacy of L1/L2 minimization on image gradients for enhanced sparsity.
- To compare L1/L2 gradient regularization against traditional L1 total variation.
- To demonstrate improvements in image recovery applications.
Main Methods:
- Development of a specific splitting scheme for numerical analysis.
- Application of the alternating direction method of multipliers (ADMM).
- Convergence analysis (subsequential and global) of the ADMM under specific conditions.
Main Results:
- Demonstrated visible improvements of L1/L2 over L1 and other nonconvex regularizations.
- Successful application in image recovery from low-frequency measurements.
- Validated effectiveness in medical imaging for MRI and CT reconstruction.
Conclusions:
- L1/L2 gradient regularization offers superior performance compared to L1 total variation for image recovery.
- Empirical evidence supports the advantage of L1/L2 for piecewise constant signal recovery.
- The proposed method shows promise for future advancements in imaging applications.
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